MULTIPLE PERIODIC SOLUTIONS OF ASYMPTOTICALLY LINEAR NONAUTONOMOUS HAMILTONIAN SYSTEMS

This paper studies the existence of periodic solutions for nonautonomous asymptoticallylinear Hamiltonian systems.By using the Z_p index theory some multiplicity results fornonautonomous systems are given,which generalize some results for autonomous systemsdue to Amman and Zenhder.     

Symmetries,morse polynomials and applications to bifurcation problems

In this paper,utilizing equivariant Mose theory for isolated critical orbits developed in [16] we study the versions of Morse identities under group actions,i.e.the global property of equivariant Morse theory. Combining the concept of critical multiplicity we give a lower bound of the critical multiplicities for a class of invariant functionals.As applications we study the bifuractions of a class of equivariant potential operators and generalize some results due to Benci and Pacella, Chang Rabinowitz, etc.This research was supported in part by the National Postdoctoral Science Fund.     

Homoclinic orbits for first order Hamiltonian systems with some twist conditions

In this paper, we study the nonperiodic first-order Hamiltonian system ù = J L(t)u +J H(t, u), where H ∈ C1(R × R2n). With some assumptions on L, the corresponding Hamiltonian operator has only discrete spectrum. By using the index theory for self-adjoint operator equation,we establish the existence of multiple homoclinic orbits for the asymptotically quadratic nonlinearty satisfying some twist conditions between infinity and origin.     

EXISTENCE OF MULTIPLE PERIODIC SOLUTIONS TO A SECOND-ORDER NONLINEAR DIFFERENCE SYSTEM

In this paper, by the critical point theory, a sufficient condition for the existence of multiple periodic solutions to a nonlinear second order difference system is obtained. An illustrative example is given. Our results improve and generalize some known ones.     

Homoclinic orbits for a class of first-order nonperiodic asymptotically quadratic Hamiltonian systems with spectrum point zero

In this paper, we study the existence and multiplicity of homoclinic orbits for a class of first-order nonperiodic Hamiltonian systems. By applying two recent critical point theorems for strongly indefinite functionals, we give some new criteria to guarantee that Hamiltonian systems with asymptotically quadratic terms and spectrum point zero have at least one and a finite number of pairs of homoclinic orbits under some adequate conditions, respectively.     

Infinitely many homoclinic orbits for a class of second‐order damped differential equations

We investigate the existence and multiplicity of homoclinic orbits for the second‐order damped differential equations For Equation 1 where L(t) and W(t,u) are neither autonomous nor periodic in t. Under certain assumptions on g, L, and W, we get infinitely many homoclinic orbits for superquadratic, subquadratic and concave–convex nonlinearities cases by using fountain theorem and dual fountain theorem in critical point theory. These results generalize and improve some existing results in the literature. Copyright © 2015 JohnWiley & Sons, Ltd.     

Multiple solutions of generalized asymptotical linear Hamiltonian systems satisfying Sturm-Liouville boundary conditions

This paper consider the multiple solutions for even Hamiltonian systems satisfying Sturm-Liouville boundary conditions. The gradient of Hamiltonian function is generalized asymptotically linear. The solutions obtained are shown to coincide with the critical points of a dual functional. Thanks to the index theory for linear Hamiltonian systems by Dong (2010) [1], we find critical points of this dual functional by verifying the assumptions of a lemma about multiple critical points given by Chang (1993) [2].     

Global dynamic behavior of a predator–prey model under ratio-dependent state impulsive control

This paper studies the global dynamic behavior of a prey–predator model with square root functional response under ratio-dependent state impulsive control strategy. It is shown that the boundary equilibrium point of the controlled system is globally asymptotically stable. An order-k periodic orbit is obtained by employing the Brouwer’s fixed point theorem. Furthermore, the critical values are determined for the existence of orbitally asymptotically stable order-1 and order-2 periodic orbits in finite time. These critical values play an important role in determining different kinds of order-k periodic orbits and can also be used for designing the control parameters to obtain the desirable dynamic behavior of the controlled prey–predator system. Moreover, it is found that the local equilibrium point is also globally asymptotically stable under the control strategy. Numerical examples are provided to validate the effectiveness and feasibility of the theoretical results.     

Monotonicity and existence of periodic orbits for projected dynamical systems on Hilbert spaces

We present here results about the existence of periodic orbits for projected dynamical systems (PDS) under Minty-Browder monotonicity conditions. The results are formulated in the general context of a Hilbert space of arbitrary (finite or infinite) dimension. The existence of periodic orbits for such PDS is deduced by means of nonlinear analysis, using a fixed point approach. It is also shown how occurrence of periodic orbits is intimately related to that of critical points (equilibria) of a PDS in certain cases.

     

Morse indices of critical manifolds generated by min-max methods with compact lie group actions and applications

We derive some upper and lower bounds for Morse indices of critical manifolds generated by min-max principles for functionals invariant under a general compact Lie group or a finite group action. The results generalize the similar results in the nonequivariant (no group action) case. In doing so, we also generalize the extension theorem of Dugundji type in the nonequivariant case to the equivariant (group action) case. As an application, we obtain a precise growth estimate for the whole sequence of critical values given by the min-max procedure for some superquadratic second-order differential equations. It is well-known that this growth estimate is crucial in showing the existence of multiple solutions of some superquadratic perturbed Hamiltonian systems and equations.     

Periodic solutions of asymptotically linear Hamiltonian systems

We prove existence and multiplicity results for periodic solutions of time dependent and time independent Hamiltonian equations, which are assumed to be asymptotically linear. The periodic solutions are found as critical points of a variational problem in a real Hilbert space. By means of a saddle point reduction this problem is reduced to the problem of finding critical points of a function defined on a finite dimensional subspace. The critical points are then found using generalized Morse theory and minimax arguments.     

Proximal point algorithms for solving convex minimization problem and common fixed points problem of asymptotically quasi-nonexpansive mappings in CAT(0) spaces with convergence analysis

In this paper, we introduce the modified proximal point algorithm for common fixed points of asymptotically quasi-nonexpansive mappings in CAT(0) spaces and also prove some convergence theorems of the proposed algorithm to a common fixed point of asymptotically quasi-nonexpansive mappings and a minimizer of a convex function. The main results in this paper improve and generalize the corresponding results given by some authors. Moreover, we then give numerical examples to illustrate and show efficiency of the proposed algorithm for supporting our main results.     

Homoclinic Orbits for a Class of Fractional Hamiltonian Systems via Variational Methods

This paper is devoted to the existence and multiplicity of homoclinic orbits for a class of fractional-order Hamiltonian systems with left and right Liouville–Weyl fractional derivatives. Here, we present a new approach via variational methods and critical point theory to obtain sufficient conditions under which the Hamiltonian system has at least one homoclinic orbit or multiple homoclinic orbits. Some results are new even for second-order Hamiltonian systems.     

Existence and multiplicity of periodic solutions for some second order differential systems with p(t)-Laplacian

In this paper, we deal with the existence and multiplicity of periodic solutions for the p(t)-Laplacian Hamiltonian system. Some new existence theorems are obtained by using the least action principle and minmax methods in critical point theory, and our results generalize and improve some existence theorems.     

Multiple periodic solutions for a class of nonlinear nonautonomous wave equations

In this paper, by means of ourZ _p index theory developed recently we study the existence of multiple periodic solutions for asymptotically linear nonautonomous wave equations. All previous known results rely either on the oddness of nonlinear terms, or on autonomous systems, and the best result for the general case is the existence of two nontrivial periodic solutions (Amann, Zehnder, K. C. Chang, S. P. Wu, S. J. Li, Z. Q. Wang). In this paper, under the assumption that the nonlinear term isT/p periodic we discuss multiple periodic solutions of nonautonomous systems and generalize a series of previous results.This research was supported in part by the National Postdoctoral Science Fund.     

EXISTENCE AND UNIQUENESS OF NON－ZERO－HOMOTOPIC PERIODIC ORBITS FOR A CYLINDER SYSTEM

ＥＸＩＳＴＥＮＣＥＡＮＤＵＮＩＱＵＥＮＥＳＳＯＦＮＯＮ－ＺＥＲＯ－ＨＯＭＯＴＯＰＩＣＰＥＲＩＯＤＩＣＯＲＢＩＴＳＦＯＲＡＣＹＬＩＮＤＥＲＳＹＳＴＥＭＨａｎＭａｏａｎＷａｎｇＸｉａｎｇｒｏｎｇＷａｎｇＣｈａｏｙａｎｇ＆ＷａｎｇＺｈｅｎｇ（Ｓｈａｎｄｏｎ...     

一类二阶常微分方程无穷多点边值问题多个正解的存在性

研究了一类二阶非线性常微分方程无穷多点边值问题多个正解的存在性,运用不动点指数理论及Holder不等式得到了方程至少存在两个正解的若干充分条件,推广和改进了相关文献的结果.     

Periodic solutions of non-autonomous second order systems with (q(t), p(t))-Laplacian

Using the least action principle in critical point theory we obtain some existence results of periodic solutions for (q(t), p(t))-Laplacian systems which generalize some existence results.     

一类二阶多点边值问题正解的存在性

利用锥上不动点定理讨论了二阶常微分方程组多点边值问题正解的存在性，所得结论推广了最近的一些结果．     

Foliations of asymptotically flat manifolds by surfaces of Willmore type

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch–Hawking mass. Thus our result has applications in the theory of general relativity.